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8). 4, pp. 3. 6 6 We recall that the best constant Mp is tan if 1 < p 5 2 and cot if 2 5 p < oo (see Pichorides [1972]). For p in ]I,oo[,this number will be called M. Rieez’8 conetant and will be denoted by Mp. 4) Deflnition. Let p be a number in [l,cm[, and let f be in Lp(G). For every positive integer n, the function is defined for palmost all z in GI2. The function HL f is called the nth truncated HiZbert fran8form off on G. The mazimal Hilbert tran8form off on G is defined for galmost all 2 bY (ii) Combining the results of Calder6n 119681 and the properties of the classical Hilbert transform on R, we obtain the basic properties of the Hilbert transform on G.

The relations (5) and (3) show that a * y<0 22 N. Asmar and E. Hewitt for every y in S n (-P). This establishes (1) and (2), and completes the proof. 13) Theorem. Let P be an order in Z '. There is a sequence of Archimedean orders PI,Pa,. . such tbat Proof. We need only show that KP, c P c h P , . 11) with S = C, to obtain an Archimedean order P, such that (3) P, n C, = P n C,. For positive integers n and N with n 2 N, use (3) to infer that P, n CN = PNn C N . For x in P, let N be a positive integer such that x is in CN n P.

That is, the set P is nondense in R" x F x {g}. 0 We next construct a useful family of orders. 8) Theorem. Let H be a countable discrete torsion-free Abelian group. Let L he a nonzero continuous real-valued homomorphism on Ra and let a be a real-valued homomorphism on H (a may be identically 0). Consider the mapping T defined on R" x F x H bY (4 +, f; u ) = L(x) - and suppose that F # (0) or that a > 1. Then T is a continuous real-valued homomorphism, and there is a nondense order P on R" x F x H such that (ii) 7-1()0,co[) 5 P s; r - ' ( [ o , w [ ) .